R?ssler U. Solid State Theory: An Introduction

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4.3 Occupation Number Representation for Electrons 95

annihilated, while for the second equation it is N + 1 due to the addition of

one particle to |Ψ.

The meaning of occupation numb er representation can be also illustrated

by applying c

†

to the N-particle state |Ψ. We ﬁnd

†

|Ψ =(−1)

†

|...n

− 1 ... (4.71)

=(−1)

2ν

(1 − (n

− 1))|...n

... (4.72)



|Ψ n

0 n



(4.73)

or in short

†

|Ψ = n

|Ψ . (4.74)

Thus the eigenvalue of c

†

is the occupation number n

of the single-

particle state with quantum numbers α

in |Ψ and

N =



†



ˆn

(4.75)

is the number operator with the eigenvalue equation

N|Ψ = N |Ψ.

The occupation number or Fock representation allows one to formulate

the system Hamiltonian in terms of creation and annihilation operators. We

give here only the rules for how the one-(

) and two-particle (

)termsof

the Hamiltonian (or any other operator) have to b e rewritten and refer for

details to the Appendix (

A.6):

→



ψ

|ψ

c

†

(4.76)

→



ijkl

ψ

(1)

(2)

|ψ

(1)

(2)

c

†

. (4.77)

These expressions have an obvious meaning: The one-particle operator

(depending on the co o rdinates of only one particle) replaces a particle with

quantum numbers α

by a particle with α

, the two-particle or interaction

operator r emoves ﬁrst a particle in the state α

andthenanotheronein

the state α

and adds particles in the states α

and α

.Theweightofthe

operators are matrix elements of the original operators taken with the single-

particle wave functions ψ

of an orthonormal set. The upper index (1) and

(2) in the matrix element of the two-particle operator indicates the pairs of

wave functions which have the same coordinates when integrating in p osition

representation.

This concept is now applied to the system of free electrons for which

†

kσ

′

} = δ

′

σσ

′

(4.78)

96 4 The Free Electron Gas

and

N =



kσ

ˆn

kσ

, ˆn

kσ

= c

†

kσ

,ψ

kσ

(x)=

√

ik·r

(s) , (4.79)

where χ

(s) is a Pauli spinor and s denotes the spin variable. The Hamilto-

nian for the jellium model of free electrons (with the jellium term H

from

Problem 4.1)

jell



8πε



k,l

k=l

− r

+ H

, (4.80)

can be rewritten (by making use of (

4.76)and(4.77)) with



→



kσ

¯h

†

kσ

(4.81)

and



k,l

k=l

4πε

− r

→ψ

4πε

− r

|ψ

′

 (4.82)

= δ

′



′

−k

(4.83)

where v

= e

/ε

is the Fourier transform of the Coulomb p otential (see

Appendix). With the replacements σ

= σ

′

= σ, σ

= σ

′

= σ

′

and k

′

→ p,

′

→ q we ﬁnd

jell



kσ

¯h

†

kσ



p,q,k

σσ

′

†

p+kσ

†

q−kσ

′

qσ

′

pσ

+ H

. (4.84)

The interaction term becomes divergent for k → 0. However, this diver-

gence is exactly compensated by the jellium term for which we obtained as

solution of Problem 4.1 H

= −

while



p,qσσ

′

†

pσ

†

qσ

′

qσ

′

pσ



pσ

ˆn

pσ



qσ

′

ˆn

qσ

′

. Thus we arrive at the Hamiltonian for free

electrons in the jellium model

jell



kσ

¯h

†

kσ



p,q,k=0

σσ

′

†

p+kσ

†

q−kσ

′

qσ

′

pσ

. (4.85)

Its ﬁrst term accounts for the single-particle energy of free noninteracting

electrons, while the second term describes the electron–electron interaction.

The interaction term is visualized in Fig. 4.10 as a Feynman diagram with the

solid lines r e presenting incoming and outgoing electrons with their respective

4.4 Hartree–Fock Approximation 97

q-kσ¢

qσ¢

p+kσ

pσ

Fig. 4.10. Feynman diagram of the electron–electron in teraction. The solid lines

represent propagating electrons, the dashed line the Coulomb potential

wave vector and spin and the dashed line symbolizes the Coulomb interaction.

This Hamiltonian does not only apply to electron systems in vacuum, but

also to those realized in matter (see Table 4.1), however,by replacing the free

electron mass by an eﬀective mass and screening the Coulomb interaction

with the dielectric constant of the matter. Moreover it applies also to electron

systems in energy bands, however, with a matrix element between full Blo ch

states (see Chaps.

5 to 7). In the following sections we develop the concepts

for approximate solutions of the ground state problem for the jellium model.

4.4 Hartree–Fock Approximation

In Sect.

4.1, we have characterized the ground state of the noninteracting

electron system (at T = 0 K) as the ﬁlled Fermi sphere. The corresponding

N-particle wave function, that takes into account the Pauli principle, is a

Slater determinant composed of the wave functions of occupied single-particle

states, i.e., plane waves with |k|≤k

but it can also be obtained by applying

creation operators to the fermion vacuum to ﬁll the Fermi sphere. Let us

denote this ground state by |Ψ

. We may now calculate the expectation value

of H

jell

from (

4.85)with|Ψ

 giving the ground state energy in Hartree

–Fock

(HF) approximation

= Ψ

jell

|Ψ

 =



kσ

Ψ

¯h

†

kσ

|Ψ





k=0

q,p,σσ

′

Ψ

†

p+kσ

†

q−kσ

′

qσ

′

pσ

|Ψ

. (4.86)

The ﬁrst term reproduces the ground state energy

of the noninteracting

electron gas obtained in the Sommerfeld model (Sect.

4.1). The second term

is the ﬁrst order correction due to the Coulomb interaction, which is treated

Douglas Rayner Hartree 1897–1958

98 4 The Free Electron Gas

direct exchange

Fig. 4.11. Illustration of the direct and exchange terms in the Fermi sphere

here as a perturbation. Before we discuss the quality of this approximation,

this correction has to be calculated. Two contributions, depicted in Fig. 4.11

can be distinguished:

1. Proceeding from right to left the two annihilation operators remove the

electrons with pσ and qσ

′

from the ﬁlled Fermi sphere (|Ψ

) and the two cre-

ation operators add electrons at p + kσ and q −kσ

′

, respectively. In order to

obtain the ﬁlled Fermi sphere (or Ψ

|) k has to be zero. However, this term is

excluded from the sum over k in the interaction term as a result of the jellium

model and hence does not modify the ground state energy of the Sommer-

feld model. It should be mentioned, however, that this so-called direct term

describes the electrostatic interaction energy for a system with charge den-

sity n (see Problem 4.1), which becomes important for inhomogenous electron

systems (see Chaps.

5 and 6) and deﬁnes the Hartr ee approximation.

2. As before, the two annihilation op erators remove two electrons from the

Fermi sphere, while the two creation operators put them back but now with

interchanged wave vectors. This exchange within the Fermi sphere is possible

only for electrons having the same spin (σ = σ

′

) and whose wave vectors

diﬀer by k = q − p (see Fig. 4.11). This exchange term results from the Pauli

principle and does not have a classical analogue (in contrast with the direct

term). The Hartr ee–Fock approximation is also understo od to have considered

the exchange contribution.

The exchange term is evaluated under the constraints of the exchange

process (k = q − p,σ= σ

′

) giving

= E



σ,q,p

q=p

2ε

V |q − p|

Ψ

†

qσ

†

pσ

qσ

pσ

|Ψ

 (4.87)

with E

bei ng the ground state energy in the Hartree approximation. Because

of p = q the operators, c

†

pσ

and c

qσ

can be interchanged with a sign change

and the four-operator expression becomes a product of number operators or

their r espective eigenvalues, the occupation factors n

qσ

pσ

. Thus one ﬁnds

4.4 Hartree–Fock Approximation 99

= E

−



σ,q,p

q=p

2ε

V |q − p|

qσ

pσ

. (4.88)

Taking E

from (

4.16) and after summing over σ and considering the

o ccupation factors, gives

−



q,p,q=p

|q|,|p|≤k

V |q − p|

. (4.89)

The evaluation of the sums is explicitly carried out in the Appendix (

A.5). It

gives

exch

= −

16π

. (4.90)

and leads to the important result of this calculation, the ground state energy

of the electron gas in HF approximation

¯h

−

16π

. (4.91)

Due to the negative contribution of the exchange interaction it is lower than

the result of the Sommerfeld model. In the formulation with the dimensionless

density par ameter r

deﬁned in Sect.

4.1 it reads

= N

9π

2/3

−

4π

9π

1/3

Ry. (4.92)

As mean value energy per electron the result is written in the form



2.2099

−

0.916



Ry. (4.93)

It is exp ressed in terms of inverse powers of the density parameter r

,which

for the same electron density n can have quite diﬀerent values depending on

the material (see Table 4.1) in which it is realized (Problem 4.6).

In or d er to go beyond the HF approximation (which is a ﬁrst order per-

turbation calculation) one may consider the second order correction becausse

of the Coulomb interaction. This is formulated in Problem 4.7. As it will turn

out, this term is logarithmically divergent. Thus higher order perturbation

calculation seems not to be the proper way to treat Coulomb interaction and

special many-body techniques are required to improve the result. But we can

take from (

4.93) the message that the HF approximation works well in the

high density limit (r

≪ 1).

At this point, two considerations about the HF approximation can be

made. The ﬁrst one aims at an approximate formulation of H

jell

(

4.85)asan

eﬀective single-particle operator

100 4 The Free Electron Gas

jell

≃



kσ

†

kσ

(4.94)

with single-particle energi es ǫ

kσ

, which besides the kinetic energy of free par-

ticles contain a contribution from the Coulomb interaction according to the

HF approximation. A systematic way to calculate ǫ

kσ

is the formulation of

the equation of motion for the time-dependent fermion operator

†

kσ

i¯h

jell

†

kσ

. (4.95)

The commutator of c

kσ

with the interaction term generates terms consisting

of three fermion op erators. Here the HF approximation means to replace these

terms by the product of an expectation value of the number operator and the

remaining fermion operator. This classiﬁes the HF approximation as a mean

ﬁeld approximation, see Sect.

6.5. The result takes the form

†

kσ

i¯h

kσ

†

kσ

(4.96)

with the single-particle energy

kσ

¯h

−



q=k

|q|≤k

|q − k|

. (4.97)

The evaluation of the sum over q in the Fermi sphere can be performed as an

integral in polar coordinates with ϑ being the angle between q and k



qk

|q|≤k

|q − k|

(2π)

2π



dqq



−1

dcosϑ

+ k

− 2kq cos ϑ

(2π)



(q − k)

(q + k)

. (4.98)

The last integral can be found in an integral table. Finally, we obtain for the

single-particle energy in HF approximation

kσ

¯h

−

4π

− k

2kk

+ k

− k

. (4.99)

Aplotofǫ

kσ

is shown in Fig. 4.12 for a particular choice of the electron density

n in reduced units together with the free electron dispersion.

As for the ground state energy, we ﬁnd for the single-particle energies in the

HF approximation a lowering with respect to the free particle energy. Taking

the sum over these single particle energies in the Fermi sphere leads back to

the result of (

4.93). Replacing the Hamiltonian of the jellium model (4.85)as

indicatedin(

4.94) corresponds to describing the interacting free electrons as

4.4 Hartree–Fock Approximation 101

Band width

0.5

1.5

(k/k

)

–1

–2

ε/ε°

Fig. 4.12. Single-particle HF energy (solid line)forr

= 4 in comparison with free

particle energy (dashed line), after [

12]

noninteracting quasi-particles, whose single-particle energies incorporate part

of the Coulomb interaction. This quasi-particle con cept is frequently used in

many-body theory.

The second consideration concerns the charge density connected with

the exchange interaction. For this purpose, we may rewrite the exchange

correction with the Fourier transform of the Coulomb interaction as

−



q=k

|q|≤k

|q − k|

= −

4πε



q=k

|q|≤k



i(q−k)·r

= −



eρ

(r)

4πε

r. (4.100)

The last expression corresponds to the potential energy of an electron at r =0

in a charge density ρ

(r) (the HF or exchange charge density) resulting from

all electrons in the Fermi sphere having the same spin:

(r)=



q=k

|q|≤k

i(q−k)·r

−ik·r



q=k

|q|≤k

iq·r

. (4.101)

By performing the sum over q as integral over the Fermi sphere in polar

coordinates one obtains

(r)=

(2π)

−ik·r

(sin (k

r) − k

r cos (k

r)) . (4.102)

The mean value of ρ

(r) over the Fermi sphere gives the averaged HF charge

density

102 4 The Free Electron Gas

g(r)

1/2

01234

Fermi or exchange hole

Fig. 4.13. Pair–distribution function g(r)=1−ρ

(r)/en showing the depression

in the vicinity of r = 0 known as Fermi or exchange hole

(r)=



|k|≤k

(r)

(sin(k

r) − k

r cos(k

r))

. (4.103)

The limits

(r →∞)=0 and ρ

(r → 0) =

en (4.104)

indicate the modiﬁcation of the constant charge density −en of the nonin-

teracting electron gas due to the exchange correction. In the vicinity of an

electron with given spin, the charge density is reduced by 50% correspo n din g

to the interaction with all electrons having the same spin. With increasing

r, the density approaches the value of the nonintera cting electron system.

This r esult can be expressed in terms of the pair–distribution function (see

Sect.

1.3)

g(r)=1−ρ

(r)/en (4.105)

for free electrons in the HF approximation, which is depicted in Fig. 4.13.

It shows a depression around r =0whichistheFermi or exchange hole

and visualizes the correlation due to the exchange interaction or the Pauli

principle.

4.5 The Dielectric Function

The dielectric function is the response of the system (here the homogeneous

electron system) to a perturbation by an electromagnetic ﬁeld. It is conve-

nient to consider this perturbation as having the characteristic space and

4.5 The Dielectric Function 103

time dependence given by exp(i(q · r − ωt)) typical for a Fourier component

with q and ω. One can distinguish this perturbation caused by a moving

charge (as in a particle scattering process) from that of an electromagnetic

wave. The latter case, leading to the transverse dielectric function has been

the subject of Sect.

3.5. Here we consider the former case: T h e moving charge

is connected with a longitudinal ﬁeld (directed parallel to the momentum of

the moving particle) and the response is the longitudinal dielectric function

ε(q,ω). For ω = 0 this case includes also the response of the electron sys-

tem to a static charge placed into the e lectron system: The electrons will

arrange around this static charge due to Coulomb attraction (if the static

charge is positive) or repulsion (if it is negative), thus increasing or reducing

the otherwise homogeneous density. This eﬀect is known as static screening.

In Sect.

2.5, the inverse dielectric function has been introduced already in

an exact formulation for an arbitrary system as density–density correlation

function between density and number ﬂuctuations. Here we want to derive the

inverse dielectric function for the homogeneous electron gas. Starting point is

the electron system described by H

jell

and the external perturbation by V

ext

H = H

jell

+ V

ext

. (4.106)

Having in mind the results of Sect.

2.5, it is advantageous to use the for-

mulation of H

jell

in terms of number ﬂuctuations



pσ

†

p+kσ

pσ

(see

Problem 4.8)

jell



kσ

¯h

†

kσ



k=0



†

− N



(4.107)

and write the perturbation as

ext

= −v

ext

iq·r−iωt

= −v

−q

ext

−iωt

(4.108)

where we have replaced exp(iq · r)by

−q

The response of the homogeneous electron system to a perturbation by an

external charge will be a deviation from the homogeneous density in terms

of numb er (or density) ﬂuctuations. For V

ext

as given in (

4.108) the induced

number ﬂuctuations are described by

N(r,t)=

e

iq·r−iωt

. (4.109)

In the context of Poisson’s equation, the external test charge eN

ext

and

the induced charge (density) ﬂuctuation eN (r,t) play the role of free and

polarization charges, respectively, or in other words: eN

ext

/V is the source of

the dielectric displacement ﬁeld D while both eN

ext

and eN (r ,t) determine

the electric ﬁeld E, both ﬁelds being connected by the dielectric function:

D(q,ω)=ε

ε(q,ω)E(q,ω). Thus, as in Sect. 2.5 we may write

104 4 The Free Electron Gas

iq · E(q,ω)=

Vε



ext

+ 





(4.110)

iq · D(q,ω)=

ext

(4.111)

and obtain from these equations the relation

ε(q,ω)

=1+





ext

. (4.112)

From (

4.108)and(4.109)weidentify

−q

and

as the operators

B and

respectively, of the response formalism, to write the exact expression for the

inverse longitudinal dielectric constant as (see (

2.80))

ε(q,ω)

= 1 + lim

Γ →0

¯h



∞

dτe

iωτ−Γτ

[

(τ),

−q

(0)]

(4.113)

where the damping factor regularizes the integral.

The thermal expectation value under the integral is to be evaluated with

the eigenstates of the Hamiltonian H

jell

of the unperturbed system. As these

are not known, this can be done only approximately. For T =0Kthethermal

expectation value reduces to the expectation value of the ground state |Ψ



(see Problem 2.2) and we may write

Ψ

(τ),

−q

(0)]|Ψ

 =



Ψ

(τ)|Ψ

Ψ

−q

(0)|Ψ



−Ψ

−q

(0)|Ψ

Ψ

(τ)|Ψ



. (4.114)

The matrix elements can b e evaluated by making use of

(τ)=e

¯h

jell

−

¯h

jell

and

−q

†

(4.115)

to yield ¯hω

= E

− E

Ψ

(τ),

−q

(0)]|Ψ

 =



|Ψ

|Ψ

|

(

−iω

− e

iω

)

. (4.116)

Thus we obtain the exact expression for the longitudinal dielectric function

at T =0K

ε(q,ω)

= 1 + lim

Γ →0

¯h



|Ψ

|Ψ

|



+ ω +iΓ

− ω − iΓ



. (4.117)

Formally, this result is the same as for χ(ω) derived in Sect.

3.5 for a trans-

verse perturbation, where however the matrix elements are those of the dipole

operator while here they are those of the operator of numb er ﬂuctuations.